A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Hitting time of a cone involving Brownian motion

I don't understand the following when reading a proof: Let $B$ be a standard Brownian motion (in $\mathbb{R}^d$) and $\{ \mathcal{F}_t \}$ be the filtration generated by $B$. Let $C$ be a cone in ...
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Density of this quantity for a Geometric Brownian Motion?

If we define $X_T = X_t e^{(\mu-\frac{1}{2}\sigma^2 ) (T-t) + \sigma W_{T-t}}$ where $W_{T-t}$ is a classical weiner process. How would we go about deriving the density and expectation for $X_{max} - ...
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34 views

Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [on hold]

Solve the following stochastic differential equations $ dX_t = \frac{1}{2 X_t} dt + dB_t$ or equivalently with a transformation $Y_t = X_t^2$ $ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
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Joint distribution of arrival times in Poisson process

I need to compute the following joint distribution in a Poisson process: $f_{S_A S_{A+B}}(t_1, t_2), t_2\ge t_1$ $S_A$ and $S_{A+B}$ are the arrival epochs of the $A^{th}$ and ${A+B}^{th}$ arrivals ...
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Empirical characterization of the Brownian Motion

A well-known characterization of the Brownian Motion says that it is the only continuous process $X_t$ (defined on $[0,\infty)$) such that $P(X_0=0)=1$, $E[(X_{t+s}-X_t)^2|X_t]=s$, ...
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14 views

Expected value of integrals of a gaussian process

I have limited knowledge of the theory of stochastic processes. While working on a problem I've stumbled upon some expected values of time integrals of Gaussian stochastic processes. Before starting ...
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16 views

continuous random walks, wiener process, ito process: “snowballing” for high enough volatility?

I'm finishing a project for my ODE class and ran into some strange behavior involving a SDE (not exactly sure how to say this, but...) generated by an Ito process, using the Wiener process. I guess ...
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1answer
21 views

Variance of integrated squared wiener process

So I'm trying to figure out the mean and variance of $X = \int_{0}^{1} W^2(t) dt $ where $W$ is the Wiener process. The mean I've worked out easily to be $\frac{\sigma^2}{2}$ but I'm having ...
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Are these transient or recurrent states in a Markov chain?

I have the following transition matrix for a Markov chain with states $A, B, C, D, E$ $ \left| \begin{array}{ccc} 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & ...
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20 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
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How to determine the probability density function, ${f_{\dot X}}\left( {\dot x} \right)$, for the derivative process of a stochastic process?

I would like to calculate the up-crossing rate ($\nu _a^ + $) for a stationary stochastic process, $X(t)$, given by the probability distribution function of its 'intensity', ${f_X}\left( x \right)$, ...
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23 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution ...
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33 views

I can't understand this difference equation step

I am working on birth-death processes and I can't understand a step that is taken in a proof. The mean of a process is defined as $$\mu(t) = \sum_{n=1}^{\infty}np_n(t)$$ At certain stage in the ...
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43 views

Probability of ultimate extinction? Need to show that an infinite series is less than $1$

I have the following probability generating function for a branching process - $$G_n(s) = \frac{n}{n+1} + \sum_{r=1}^{\infty}\frac{n^{r-1}}{(n+1)^{r+1}}s^r$$ It says in a book that extinction is ...
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8 views

Do we need Feller condition if volatility jumps?

Consider the SDE: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} \end{equation} It describes a process $v_t$ which is a strictly positive if the drift is stronger enough, i.e. ...
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23 views

Brownian brigde, brownian motion and independence.

Let $\{W(t)\}_{0 \le t \le 1}$ a Brownian motion. Then $\{B(t)\}_{0 \le t \le 1}$ with $B(t)=W(t)-tW(1)$ is a Brownian brigde. My goal is to prove that $B(t)$ and $W(1)$ are independent. Since their ...
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22 views

Proving an equality for a Markov Chain

Let $X_n$ be an irreducible Markov chain taking values from the natural numbers (including $0$). Let $g,f$ be functions with $\mathbb{N}$ as domain (including $0$) such that $f = g + Pf$, where $P$ is ...
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13 views

Looking for solutions manual: Probability and Stochastic Processes for Engineers [on hold]

My first posting to this community. I am an engineer. I am trying to teach myself the elements of Stochastic Processes. I found the book "Probability and Stochastic Processes for Engineers" By C. ...
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17 views

Continuous Time Markov Chains Example

I have taken a class on discrete time markov chains which i clearly understood the properties, calculating probabilities and the steady state distribution. However, After progressing into the ...
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10 views

detailed balance condition for coupled Langevin equation

Suppose $a$ and $m$ are real variables and they satisfy the following two coupled Langevin equations: $$ \dot{a}=F_a(a,m)+\eta_a(t);\quad\dot{m}=F_m(a,m)+\eta_m(t) $$ where $\eta_a$ and $\eta_m$ are ...
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two definition of the Poisson process

I read the definition of Poisson process in Shereve's "Stochastic Analysis" which is constructed explicitly: (A) step 1:construct iid exponential r.v.$\tau_i$ with parameter $\lambda$ step 2:define ...
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32 views

Measurability of the points of (strict) increase for Stochastic Process

Given a background space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ , I'm considering a stochastic process $X:=(X_{t})_{t\geq0}$ with distribution $X(\mathbb{P})$ on ...
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39 views

Bounding an expected hitting time

Consider a stochastic differential equation: $$dX_t = dW_t + \sin(X_t) dt, \, X_0 = x$$ where $W_t$ is a Wiener process. Define $$\tau_1 = \inf \{ t : X_t \in 2 \pi \mathbb{Z} \} \\ \tau_2 = \inf ...
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1answer
22 views

Can a chain with repeated nodes still be considered a Markov chain?

The well-known Markov Property is that $$P(X_n = i | X_{n-1} = k_1, \dots, X_{n-j} = k_n ) = P(X_n = i | X_{n-1} = k_1) $$ Suppose we lay out some stochastic model in the following transition ...
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10 views

Is $X(u,t)=\int^t_0V(u,\alpha)d\alpha$ wide-sense stationary?

I got a problem. The velocity of a particle in the X-direction is $V(u,t)$, where $V(u,t)=\pm V.$ $V(u,t)$ changes direction on average of $\lambda$ times per seconds, according to a Poisson law, ...
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31 views

Master equation of chemical reaction

I have about the construction of master equation for chemical reaction i.e. I have to construct differential equations for the probability mass function for the number of particles A, B and C. When ...
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1answer
35 views

Uniformly integrable martingale in a finite time horizon

Let $\{ M (t) \mid t \in [0,T] \}$ be a martingale and $\{ \tau_n \mid n = 1, 2, \ldots\}$ be an increasing sequence of stopping times such that $\tau_n \rightarrow \infty$ as $n \rightarrow \infty$. ...
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Density of $\int_{0}^{t}W'(B_{s})ds$ where $W'$ is smooth and compactly supported.

Only hints please Density of $\int_{0}^{t}W'(B_{s})ds$, where $B_{s}$ is 1-d Brownian motion. The density of $Y_{s}:=W'(B_{s})$ is $g_{Y}(y)=p_{B_{s}}((W')^{-1}(y))|\frac{d(W')^{-1}(y)}{dy}|$. How ...
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16 views

Gibbs Sampler integral computeable

here is an example of a changepoint in a poisson world with the gibbs sampler, it is an bayesian approach. the data are assumed to follow this distributions : $\begin{equation} \nonumber Y_i \sim ...
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1answer
38 views

Conditional Gambler's Ruin

I've learned about the most canonical gambler's ruin problems, but what if winning or losing on a previous turn changes the probability of winning or losing on the following turn? Say each turn I ...
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Apply Ito's formula to Bessel prosess [on hold]

Let $X_t=\sqrt{(B_t^1)^2+(B_t^2)^2+(B_t^3)^2}$ be a Bessel process (starting from 0) with respect to a 3-dimentional standard Brownian Motion $B_t=(B_t^1,B_t^2,B_t^3)$. How to apply Ito's formula to ...
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Robustness of Markov Chains

A Markov Chain on a measurable space $X$ is uniquely determined by a stochastic kernel $P$ on $X$. Let $\mathsf P_x$ denote the probability on paths generated by $P$ and the initial condition $x\in ...
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I want Find finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions [on hold]

Write the finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions. Thanks for help.
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$\chi_q=\lambda t E[Y_{1}^q],$ $\psi(u,t)=\exp[-\lambda t (1-\psi Y_{1}(u))]$ is a compound Poisson process has the second moment properties . [on hold]

Let $C(t)=\sum_{n \ge1}Y_{n}1_{\{t\ge T_{n}\}}=\sum_{n=1}^{N(t)}Y_{n}$ is compound Poisson process. Show that a compound Poisson process has the second moment properties in $E[C(t)]=\lambda tE[Y_1],$ ...
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27 views

If $E[X(t)X(s)]=t \land s $. Show that this process has independent increments [on hold]

Let $X(t), t\ge0$ be a real-valued Gaussian process with zero mean and the covariance function $\mathbb{E}\left[X(t)X(s)\right] = t \land s $. Show that this process has independent increments.
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Question about calculate expected value

Assume $X(t)$ is a Brownian motion. Find $E[X(u)X(u+v)X(u+v+w)]$, where $0<u<u+v<u+v+w$ I have an idea to solve this problem, as follows: ...
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33 views

Continuou Time Markov Chains - Poisson Distribution

Suppose $X_t$ and $Y_t$ are independent Poisson processes with parameters $\lambda_1$ and $\lambda_2$, respectively, measuring the number of calls arriving at two different phones. Let ...
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1answer
48 views

Compute almost sure limit of martingale?

Let $Y_1, Y_2, \dots$ be nonnegative i.i.d random variables with mean 1. Let $$X_n = \prod_{m \le n}Y_m$$ If $P(Y = 1) < 1$, prove that $\lim_{n->\infty}X_n = 0$ almost surely. I feel like ...
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1answer
16 views

Ito integral's zero mean

My Sto Cal prof gave a long proof for the fact that $E[\int_{0}^{t} f_s dW_s] = 0$ where W is Brownian and f is Borel x $\mathscr{F}$-measurable, adapted and satisfies some integrability condition. ...
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22 views

Learning Stochastic Processing, Modeling, and Analysis: Any Available Workbooks?

Motivation behind the question: I took the upper-level probability course at my college, and did pretty well. Most of the time throughout the class, I found myself intuitively understanding the ...
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1answer
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Asymptotic behaviour of absolute different of two independent Poisson processes

Suppose we have $X_1,X_2\sim Po(\lambda)$ ($X_1$ and $X_2$ are independent). Consider the interval $[0,1]$ with 100000 subintervals of length $\Delta=\frac{1}{100000}$. I can calculate: ...
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markov property in Durrett's textbook

Assume $B_t(\omega)=\omega(t),\omega\in (C,\mathcal{C},\mathbb{P}^x)$ is a B.M.(C is the continuous function space ,$\mathcal{C}$ is generated by the coordinate maps) In Durrett's textbook,he proved ...
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28 views

If $X^T(t)=X(t\land T)$ is said to be the process $X$ stopped at $T$. I want prove following statment

Let $X$ be a stochastic process defined on a probability space $(\omega ,\mathcal F,P)$ endowed with a filtration $(\mathcal F)_{t \ge0}$ and let $T$ , $T^\prime$ be $\mathcal F_{t}-$stopping times. ...
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1answer
14 views

Show that the expectation of a submartingale and supermartingale is an increasing and decreasing function of time, respectively.

Show that the expectation of a submartingale and supermartingale is an increasing and decreasing function of time, respectively. thanks for help
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Computing world states from uncomprehensive sensor readings

I have a real world system, which consists of items assuming different locations at different times. The state transitions are controlled by machinery in the real world, which is well understood. ...
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1answer
87 views

How to solve a discrete SIR epidemic model?

Let $(S(t), I(t), R(t))$ be a continuous time Markov chain SIR model with discrete space, where $S(t)$ stands for the number of susceptible people at time $t$; $I(t)$ stands for the number of ...
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21 views

Calculating expectation function and covariance function

Let $E_n(t)$ denote the empirical cdf based on iid uniform $u[0,1]$ random variables $U_1,...,U_n.$ The corresponding uniform empirical process $(e_n(t),0\leq t\leq 1)$ is given by ...
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probablity that $\max(X,Y)> a \min(X,Y)$

Two independent random variable $X$ and $Y$ having probability density functions uniform in the interval [0,1]. when $a \geqslant 1$, the probability that $\max(X,Y)> a \min(X,Y)$ is? (in terms of ...
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1answer
39 views

Brownian motion is almost surely continuous

Why is Brownian motion required to be almost surely continuous instead of merely continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
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1answer
27 views

Applications of Singular Functions

For our purposes here, a singular function is a continuous function such that the part which is absolutely continuous with respect to Lebesgue measure is zero. For example, the Cantor function or ...